Rotational Hypersurfaces in S3(r)R Product Space

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Geliş/Received

05.09.2016

Kabul/Accepted

14.12.2016

Doi

doi: 10.16984/saufenbilder.284219

Rotational Hypersurfaces in

S

3

(

r

)



R

Product Space

Erhan Güler

1*

_{, Ömer Kişi}

2 ABSTRACT

We consider rotational hypersurfaces in S3(r)R product space of five dimensional Euclidean space E5. We calculate the mean curvature and the Gaussian curvature, and give some results

Keywords: 5-space, rotational hypersurface, shape operator, Gaussian curvature, mean curvature

R

S

3

(

r

)



Çarpım Uzayındaki Dönel Hiperyüzeyler

ÖZ

Beş boyutlu Öklid uzayı _{E içindeki}5 _S _R ) (

3

r çarpım uzayının dönel hiperyüzeylerini ele aldık. Hiperyüzeylerin ortalama eğriliği ve Gauss eğriliğini hesapladık ve bunların bazı sonuçlarını verdik

Anahtar Kelimeler: 5-boyut, dönel hiperyüzey, şekil operatörü, Gauss eğriliği, ortalama eğrilik

1_{Bartın University, Faculty of Sciences, Department of Mathematics, Bartın - eguler@bartin.edu.tr} 2_{Bartın University, Faculty of Sciences, Department of Mathematics, Bartın - okisi@bartin.edu.tr} *_{Corresponding Author}

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Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 21, no. 3: pp. 350-355, 2017 351

1. INTRODUCTION (GİRİŞ)

When we focus on the ruled (helicoid) and rotational characters in literature, we see Bour's theorem in [2]. About helicoidal surfaces in Euclidean 3-space, do Carmo and Dajczer [4] prove that, by using a result of Bour [2], there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface.

Magid, Scharlach and Vrancken [6] introduce the affine umbilical surfaces in 4-space. Vlachos [12] consider hypersurfaces in

E

4 with harmonic mean curvature vector field. Scharlach [11] studies the affine geometry of surfaces and hypersurfaces in 4-space. Cheng and Wan [3] consider complete hypersurfaces of 4-space with constant mean curvature.

General rotational surfaces as a source of examples of surfaces in the four dimensional Euclidean space were introduced by Moore [7, 8]. Ganchev and Milousheva [5] consider the analogue of these surfaces in the Minkowski 4-space. They classify completely the minimal general rotational surfaces and the general rotational surfaces consisting of parabolic points. Arslan et al [1] study on generalized rotation surfaces in 4.

E Moruz and Munteanu [9] consider hypersurfaces in the Euclidean space

E

4 defined as the sum of a curve and a surface whose mean curvature vanishes. They call them minimal translation hypersurfaces in

E

4 and give a classification of these hypersurfaces.

We consider the rotational hypersurfaces in S3(r)R of Euclidean 5-space

E

5 in this paper. We give some basic notions of the five dimensional Euclidean geometry in Section 2. In Section 3, we give the definition of a rotational hypersurface in S3( )R

r of

E

5

.

Then we calculate the mean curvature and the Gaussian curvature of the rotational hypersurface.

2. PRELIMINARIES (ÖN HAZIRLIK)

In the next representations and definitions we inspire the three dimensional Euclidean space and the book of O’Neill [10], and then extend it to the dimension five. In this section, we will introduce the first and second fundamental forms, matrix of the shape operator

S

,

Gaussian curvature

K

and the mean curvature

H

of hypersurface

M



M

(

r

,



₁

,



₂

,



₃

)

in Euclidean 5-space

5

E

. In the rest of this work, we shall identify a vector



 with its transpose.

Definition 1. Let MM(r,₁,₂,₃) be an isometric immersion of a hypersurface

M

4 in the

E

5. The vector

product of

),

,

(

x

1

x

2

x

3

x

4

x

5

x





y





(

y

₁

,

y

₂

,

y

₃

,

y

₄

,

y

₅

),

,

(

z

1

z

2

z

3

z

4

z

5

z





w



(

w

1

,

w

2

,

w

3

,

w

4

,

w

5

)



on

E

5 is defined as follows: (1) . det 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1                     w w w w w z z z z z y y y y y x x x x x e e e e e w z y x   

Definition 2. For a hypersurface

M

(

r

,



₁

,



₂

,



₃

)

in 5-space, the first fundamental form matrix (𝑔𝑖𝑗) of

M

is

as follows (2) , I              S Q J D Q C B A J B G F D A F E and then





, 2 2 2 2 2 (3) 2 2 I det 2 2 2 2 2 2 2 2 2 2 2 2 AFJQ ABFS BJQE AGQD BFQD CFJD ABJD E GQ SE B E CJ CGD Q F GS A J A D B CS F EG                 

and the second fundamental form matrix (ℎ𝑖𝑗) of

M

is

as follows (4) II              I Z Y X Z V T P Y T N M X P M L and then





, (5) 2 2 2 2 2 2 2 II det 2 2 2 2 2 2 2 2 2 2 2 2 Y P Z M LNZ LVY NVX LIT X T NIP LTYZ VMXY MTXZ NPXZ PTXY MPYZ IMPT VI M LN                  where

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Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 21, no. 3: pp. 350-355, 2017 352 , 3  M M   _r D , 3 1   M M   J , 3 2   M M   Q , 3 3   M M   S

,

2

e

P



M

_r



T



M

12



e

,

2 2

e

V



M





Z



M

23



e

,

X



M

r3



e

,

3 1

e

Y



M





Z



M

23



e

,

I



M

33



e

,

(6) 3 2 1 3 2 1       M M M M M M M M        r r e

is the Gauss map (i.e. the unit normal vector),

"



"

means dot product, and some partial differentials that we represent are _, r r _   M M 3 1 3 1    _ _   M M .

Definition 3. Following product matrices:

, 1                          I Z Y X Z V T P Y T N M X P M L S Q J D Q C B A J B G F D A F E

gives the matrix of the shape operator

S

as follows:

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,

I

det

1

44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11



















s

S

where , 2 2 2 2 2 2 11 GLQ FMQ FJPQ BFQX AGQX CFJX BFPS AGPS CGLS CFMS BJLQ BJLQ AJMQ ABJX ABMS GPQD CGXD BMQD BJPD CJMD XD B LS B L CJ P AJ s                         , 2 2 2 2 2 2 12 FJQT GMQ FNQ BFQY AGQY BFST AGST CFJY CGMS CFNS BJMQ BJMQ AJNQ ABJY ABNS GQTD CGYD BNQD BJTD CJND YD B MS B M CJ T AJ s                         , 2 2 2 2 2 2 13 T FQ GPQ FJQV BFSV BFQZ AGSV AGQZ CFST AJQT CGPS CFJZ BJPQ BJQP ABST ABJZ GQVD BQTD CGZD BJVD CJTD ZD B PS B P CJ V AJ s                         , 2 X GQ FOQY FJQZ CGSX CFSY BJQX BFSZ AGSZ BJQX AJQY ABSY            , 2 2 2 2 2 2 21 FLQ AFQX AFPS CFLS AJLQ ABLS E MQ JPQE BQXE FPQD CJXE BPSE CMSE CFXD AMQD BLQD AMQD AJPD ABXD CJLD BPD CMD JX A MS A s                          , 2 2 2 2 2 2 22 FMQ AFQY AFST CFMS AJMQ ABMS JQTE E NQ FQTD BQYE BSTE CJYE CNSE CFYD ANQD AJTD BMQD ANQD ABYD CJMD BTD CND NS A JY A s                         , 2 2 2 2 2 23 FPQ AFSV AFQZ CFPS AJPQ ABPS OQTE JQVE FQVD BSVE BQZE CSTE CJZE AQTD AQTD CFZD BQPD AJVD CJPD ABZD BVD CTD ST A JZ A s                         , 2 2 2 2 2 2 24 X FQ CFSX AJQX AFSZ ABSX BQEI CJEI CFDI ABDI AFQI YE Q JQZE FQZD CSYE BSZE AQYD BQXD AQYD CJXD AJZD BZD CYD JI A SY A s                          , 2 2 2 2 2 2 2 31 FJLQ AFJX BFLS AGLS AFMS GQXE GPSE JMQE BJXE BMSE GLQD FMQD FJPD BFXD AGXD BJLD AJMD PE J GPD BMD QX F PS F L AJ s                        , 2 2 2 2 2 2 2 32 FJMQ AFJY BFMS AGMS AFNS GQYE GSTE JNQE BJYE FJTD BNSE GMQD FNQD BFYD AGYD BJMD AJND TE J GTD BND QY F ST F M AJ s                        , 2 2 2 2 2 2 2 33 FJQP AFST BFPS AGPS AFJZ GSVE GQZE JQTE BSTE FQTD BJZE GQPD FJVD BFZD AGZD AJTD BJPD VE J GVD BTD SV F OZ F P AJ s                        , 2 2 2 2 2 2 2 34 FJQX BFSX AGSX AFSY GQEI BJEI BFDI AGDI AFJI GSZE JQYE BSYE GQXD FQYD FJZD BJXD AJYD QI F ZE J GZD BYD SZ F X AJ s                        , 2 2 2 2 2 2 2 41 BFLQ AGLQ AFMQ AFJP ABFX CFJL ABJL GPQE CGXE BMQE BJPE CJME BFPD AGPD CGLD CFMD ABMD XE B LD B PQ F X CF GX A JM A s                       

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Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 21, no. 3: pp. 350-355, 2017 353 , 2 2 2 2 2 2 2 42 BFMQ AGMQ AFNQ AFJT ABFY CFJM ABJM GQTE CGYE BNQE BJTE BFTD AGTD CJNE CGMD CFND ABND YE B MD B QT F Y CF GY A JN A s                        , 2 2 2 2 2 2 2 43 AFQT BFPQ AGPQ AFJV CFJP ABFZ ABJP GQVE BQTE CGZE BJVE CJTE BFVD AGVD CFTD CGPD ABTD ZE B PD B QV F Z CF GZ A JT A s                        . 2 2 2 2 2 2 2 44 BFQX AGQX AFQY CFJX AFJZ ABJX CGEI ABFI GQZE BQYE CJYE BJZE CGXD CFYD BFZD AGZD ABYD EI B I CF GI A XD B QZ F JY A s                       

Definition 4. The formulas of the Gaussian and the mean curvatures are, respectively, as follow:

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,

I

det

II

det

)

(

det



S

K

and

 

,

(9)

4

1 S

tr

H



where

 

,

I

det





S

tr

. 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 FMQ FJPQ BFQX AGQX AFQY FJQP BFQX AGQX GLQ AFQY AFST CFJX BFPS AGPS AFJZ CGLS CFMS BJLQ AJMQ BJLQ AJMQ ABJX ABMS CGEI ABFI GQZE GSVE GQZE JQTE E NQ JQTE BQYE CND FQTD BQYE BSTE FQTD GPQD CJYE BJZE GQPD FJVD CNSE CGXD CFYD BFZD AGZD BMQD ANQD AJTD BMQD ANQD BJPD ABYD CJMD EI B I CF GI A VE J GVD XD B BTD QZ F SV F OZ F JY A NS A LS B L CJ P AJ                                                                       

A hypersurface

M

is minimal if

H



0

identically on

M

.

3. ROTATIONAL HYPERSURFACES (DÖNEL HİPERYÜZEYLER)

We define the rotational hypersurface in S3( )R r product space of

E

5. For an open interval IR, let

 

I :



be a curve in a plane



in

E

5, and let



be a straight line in



.

Definition 5. A rotational hypersurface in S3(r)R of

5

E

is hypersurface created by rotating a curve



around

a line



(these are called the profile curve and the axis, respectively).

We may suppose that



is the line spanned by the vector





t

1 ,

0 ,

0

. The orthogonal matrix which fixes the above vector is  ) , , (₁₂₃ Z , 1 0 0 0 0 0 cos 0 0 sin 0 sin sin cos 0 cos sin 0 sin cos sin sin sin cos cos cos sin 0 sin cos cos sin cos sin cos cos cos 3 3 3 2 2 3 2 3 2 1 2 1 1 3 2 1 3 2 1 2 1 1 3 2 1                                               

where



1

,



2

,



3



R

.

The matrix

Z

can be found by solving the following equations simultaneously;

.

1 det

,



₅



Z

ZZ

Z



t t

I

When the axis of rotation is



, there is an Euclidean transformation by which the axis is



transformed to the

5

x

-axis of

E

5. Parametrization of the profile curve is given by

 



,

0 ,



,

)

(

r



r





where



 

r : I RR is a differentiable function for all rI. So, the rotational hypersurface which is spanned by the vector



0 ,

1 

, is as follows:

t

r

,

)

(

,

)

(

)

(



₁



₂



₃

Z



₁



₂



₃



R



in 5,

E where

r



I

,



₁

,



₂

,



₃





0 ,

2 



.

Then we see the rotational hypersurface as follows:

) 10 ( . ) ( sin cos sin cos cos sin cos cos cos ) , , , ( 5 4 3 2 1 3 3 2 3 2 1 3 2 1 3 2 1                                   x x x x x r r r r r r              R

Here, we see the sphere with radius

r

: 2 2

4 1

r

x

_i i





 in R

S3(r) product space of E5. When



₂





₃



0 ,

we have rotational surface of

E

3.

Next, we obtain the mean curvature and the Gaussian curvature of the rotational hypersurface in (10).

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Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 21, no. 3: pp. 350-355, 2017 354 follow, respectively,

,

)

1 (

2 3 3 3 











r

K

and





.

1

4

3

2 / 3 2 3  













r

H

where

r



R



{

0 }

,

0 



₁

,



₂

,



₃



2 

and

 

r : I RR



is a differentiable function for all I

r .

Proof. Using the first differentials of (10) with respect to

,



1



2



3

r

we get the first quantities in (2) as follow:

.

0

0 cos

0

0 cos

cos

0

1 I

2 3 2 2 3 2 2 2 2 2











 





r





We have

,

cos

)

1 (

I

det



r

6







2 2



₂ 4



₃ where







(r

),

.

dr

d



 



Using the second differentials with respect to

r

,



₁

,



₂

,



₃

,

we have the second quantities in (4) as follow:

,

0

0 det

1 II



















c

b

a

I

where

,

cos

₂ 2 ₃ 3









r

a

,

cos

3 4 2 4









r

b

3 2 2 4

cos









r

c

and

)

1 (





The Gauss map of the rotational hypersurface (10), using (6), is

,

1 sin

cos

sin

cos

sin

cos

1

3 3 2 3 2 1 3 2 1



















W

R

e

where

_W



1 



2

.

Using (7), we get the matrix of the shape operator of the rotational hypersurface (10) as follows:

S  ❖❖ 1❖2 3/2 0 0 0 0 ❖ r 1❖2 1/2 0 0 0 0 ❖ r 1❖2 1/2 0 0 0 0 ❖ r 1❖2 1/2 .

Finally, using (8) and (9), respectively, we calculate the Gaussian curvature and the mean curvature of the rotational hypersurface (10) as follow:

,

)

1 (

I

det

II

det

)

(

det

₃ ₂ ₃ 3 











r

K

S

and





.

1

4

3

3 )

(

4

1

2 / 3 2 3  













r

tr

H

S

Corollary 1. Let

R

:

M

4





E

5 be an isometric

immersion given by (10). Then

M

4 has constant

Gaussian curvature if and only if

.

0 )

1 (

2 3 3 3



_

_

_



_





Cr

R

:

M

4





E

5be an isometric immersion given by (10) . Then

M

4 has constant mean

(6)

Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 21, no. 3: pp. 350-355, 2017 355



_r







3 

3



3 





2



16 _Cr

2



1 



2



3



0 .

R

:

M

4





E

5 be an isometric

immersion given by (10). Then

M

4 has zero Gaussian

curvature if and only if









.

1 )

(

or

,

)

(

or

,

)

(

2 1 2 1 1

c

r

c

r

c

r

c

r

c

r

















Proof. Solving the 2nd order differential eq.

K



0 ,

i.e.

,

0

3



_





we get the solutions.

R

:

M

4





E

5 be an isometric immersion given by (10). Then

M

4 has zero mean

curvature if and only if

.

1 )

(

₂ 6 1

c

r

c

dr

r









_



Proof. When we solve the 2nd order differential eq.

,

0 

H

i.e.

,

0

3











_



_



r

we get the solution.

4. SONUÇLAR (CONCLUSION)

In the present paper, we define a new kind rotational hypersurface with 4-parameters S3(r)R product space of five dimensional Euclidean space

E

5. It can be extended higher dimensions, for example S7(r)R

.

Moreover, the topic can also be transformed into the Minkowski geometry.

REFERENCES (KAYNAKÇA)

[1] K. Arslan, B. Kılıç Bayram, B. Bulca, G. Öztürk, “Generalized Rotation Surfaces in 4

E ,” Result Math. vol. 61, pp. 315–327, 2012.

[2] E. Bour, “Théorie de la déformation des surfaces,” J. de l.Êcole Imperiale Polytechnique vol. 22, no. 39, pp. 1–148, 1862.

[3] Q.M. Cheng, Q.R. Wan, “Complete hypersurfaces of

R

4 with constant mean curvature,” Monatsh. Math. vol. 118, no. 3-4, pp. 171–204, 1994. [4] M. Do Carmo, M. Dajczer, “Helicoidal surfaces

with constant mean curvature,” Tohoku Math. J. vol. 34 pp. 351–367, 1982.

[5] G. Ganchev, V. Milousheva, “General rotational surfaces in the 4-dimensional Minkowski space,” Turkish J. Math. vol. 38, pp. 883–895, 2014. [6] M. Magid, C. Scharlach, L. Vrancken, “Affine

umbilical surfaces in

R

4,” Manuscripta Math. vol. 88, pp. 275–289, 1995.

[7] C. Moore, “Surfaces of rotation in a space of four dimensions,” Ann. Math. vol. 21, pp. 81–93, 1919.

[8] C. Moore, “Rotation surfaces of constant curvature in space of four dimensions,” Bull. Amer. Math. Soc. Vol. 26, pp. 454–460, 1920. [9] M. Moruz, M.I. Munteanu, “Minimal translation

hypersurfaces in 4

E ,” J. Math. Anal. Appl. vol. 439, pp. 798–812, 2016.

[10] B. O'Neill, “Elementary Differential Geometry,” Revised second edition, Elsevier/Academic Press, Amsterdam, 2006.

[11] C. Scharlach, “Affine geometry of surfaces and hypersurfaces in _{R ,” Symposium on the}4 Differential Geometry of Submanifolds, France, pp. 251–256, 2007.

[12] Th. Vlachos, “Hypersurfaces in

E

4 with harmonic mean curvature vector field,” Math. Nachr. vol. 172, pp. 145–169, 1995.